"pull out all the stops" is a saying that means "put everything you have into the task", "use all resources", "spare no effort".
I believe it refers to organ stops. On a pipe organ, each rank of pipes is controlled by a knob that is pulled out to allow air to flow to that set of pipes. The knob is called a "stop control"; the "stop" is the rank of pipes that all have the same sound quality (flute stop, trumpet stop, reed stop, etc.) If you pull out all the stops on an organ you are using every single pipe in the organ that is available to you, all at the same time.